I've been trying to wrap my head around this for a while now, but am having trouble understanding something. This is what the textbook says:
In the first highlighted region, what does the k (arrow) f(k) mean? I've never come across this notation before, and I can't even google it because I don't know what it's called.
Also, what does it mean when it says "the class of all cycles" in the second region.
On
The $ \mapsto $ arrow is called "maps to" and it is a way to define a function (or other mapping) without giving it a name. $ x \mapsto f(x)$ is like saying there exists a function $g$ such that $g(x) = f(x) $ without the additional bookkeeping of adding a new function name. Alternatively it can be read as a lamba function $ \lambda x.f(x) $.
The "class of all cycles" means a collection of all things that have the "cycle" property. It is most likely being called a class because the collection is too large to be contained within a set.
The notation $k \mapsto f(k) $ means that an object $k $ is mapped to the image $f(k) $. This doesn't tell much, but sometimes you could see, for example, (with $x $ a real number):
$$x \mapsto x^2$$
Meaning $x $ would be mapped to its square.
As to the second part, I am not 100% sure but I think they are just referring to all cycles. "The class of all cycles" is just the set with all the cycles in it.